1. Warm Up Simplify. Solve by factoring. 1. 2. 19 3. 4. x = –4
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Warm Up
Simplify.
Solve by factoring. 1. 2. 19 3. 4.
x = –4
5. x2 + 8x + 16 = 0

2. In the previous lesson, you solved quadratic equations by isolating x2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square.
When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term.
X2 + 6x x2 – 8x + 16
Divide the coefficient of the x-term by 2, then square the result to get the constant term.

3. An expression in the form x2 + bx is not a perfect square
An expression in the form x2 + bx is not a perfect square. However, you can use the relationship shown above to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.

4. Example 1: Completing the Square
Complete the square to form a perfect square trinomial.
A. x2 + 2x +
B. x2 – 6x +
x2 + 2x + 1
x2 – 6x + 9

5. Check It Out! Example 1
Complete the square to form a perfect square trinomial.
a. x2 + 12x +
b. x2 – 5x +
x2 – 5x +
x2 + 12x + 36

6. Check It Out! Example 1
Complete the square to form a perfect square trinomial.
c. 8x + x2 +
x2 + 8x + 16

7. To solve a quadratic equation in the form x2 + bx = c, first complete the square of x2 + bx. Then you can solve using square roots.

8. Solving a Quadratic Equation by Completing the Square

9. Example 2A: Solving x2 +bx = c by Completing the Square
Solve by completing the square. Check your answer.
x2 + 16x = –15
{-15,-1}

10. Example 2B: Solving x2 +bx = c
Solve by completing the square. Write your answer in simplest radical form. Check your answer.
x2 – 4x – 6 = 0
2− 10 ,

11. Check It Out! Example 2a
Solve by completing the square. Check your answer.
x2 + 10x = –9
{-9,-1}
Plug in the greater value

12. Check It Out! Example 2b
Solve by completing the square. Write your answer in simplest radical form. Check your answer.
t2 – 8t – 5 = 0
4− 21 ,

13. Example 3A: Solving ax2 + bx = c by Completing the Square
Solve by completing the square.
–3x2 + 12x – 15 = 0
There is no real number whose square is negative, so there are no real solutions.

14. Check It Out! Example 3b
Solve by completing the square.
4t2 – 4t + 9 = 0
There is no real number whose square is negative, so there are no real solutions.

15. Example 3B: Solving ax2 + bx = c by Completing the Square
Solve by completing the square.
5x2 + 19x = 4
−4, 1 5
Plug in the negative value

16. Check It Out! Example 3a
Solve by completing the square.
3x2 – 5x – 2 = 0
− 1 3 ,2
Plug in the sum of the values as an improper fraction.

17. Example 4: Problem-Solving Application
A rectangular room has an area of 195 square feet. Its width is 2 feet shorter than its length. Find the dimensions of the room. Round to the nearest hundredth of a foot, if necessary.
Negative numbers are not reasonable for length, so x = 13 is the only solution that makes sense.
The width is 13 feet, and the length is , or 15, feet.

18. Check It Out! Example 4
An architect designs a rectangular room with an area of 400 ft2. The length is to be 8 ft longer than the width. Find the dimensions of the room. Round your answers to the nearest tenth of a foot.
The width is approximately16.4 feet, and the length is , or approximately 24.4, feet.

19. Lesson Quiz: Part I
Complete the square to form a perfect square trinomial.
1. x2 +11x +
2. x2 – 18x +
Solve by completing the square.
3. x2 – 2x – 1 = 0
4. 3x2 + 6x = 144
5. 4x x = 23 81
6, –8

20. Lesson Quiz: Part II
6. Dymond is painting a rectangular banner for a football game. She has enough paint to cover 120 ft2. She wants the length of the banner to be 7 ft longer than the width. What dimensions should Dymond use for the banner?
8 feet by 15 feet